I work in the magazine publishing world and have recently outsourced our design elements of our magazines to a third party company. We create all our files on InDesign and collaborate on these files via Dropbox. One issue we continue to struggle with is stepping on each others toes — when one person is in a file making changes and another person comes in simultaneously and makes changes then saves, over-riding the first person's saved changes. We've tried to implement steps to safeguard against these types of over-ride errors, but they do continue to happen as there are just too many people working on these files. It sounds like Microsoft products have a "Badge" to indicate another person is in the file already, does Adobe have any safeguard capabilities such as this?
InDesign does have a mechanism to avoid such access conflicts: when a document is opened, an ‘idlk’ (InDesign Lock) file is created. Unfortunately, not all network and cloud-based file systems synchronise this file because it is a temporary file; if I recall correctly, the filename begins with ‘~’ (tilde) to indicate its temporary nature.
That said, many people do seem to be successful using Dropbox with InDesign. You will find some of them, and perhaps get better advice about the matter, if you instead ask this question on the InDesign forum.
I find this happens less often if you save and share files via the Creative Cloud files because the idlk file shows up pretty quickly. However, this might be a situation where you make use of assignments in InDesign and InCopy CC, which should with your Creative Cloud subscription. It should force you to check the assignment out before making changes. Writing and copy editing software | Download free Adobe InCopy CC trial